From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 141

the sum on the left-hand side being computed with respect to the strong operator
topology ifNis infinite.
(b)Requirement (2), taking (1) into account impliesρ(0) = 0.
(c)Quantum states do exist. It is immediately proven that, in fact,ψ∈H
with||ψ||=1defines a quantum stateρψas

ρψ(P)=〈ψ,Pψ〉 P∈L(H). (2.79)

This is in nice agreement with what we already know and proves that these types
of quantum states are one-to-one with the elements ofPHas well known.
However these states do not exhaustS(H). In fact, it immediately arises from
Definition 2.3.21 that the set of the states is convex: If∑ ρ 1 ,...,ρn∈S(H)then
n
j=1pkρk∈S(H)ifpk≥^0 and

∑n
k=1pk=1. These convex combinations of
states generally do not have the formρψ.
(d)Restricting ourselves to a maximal setL 0 of pairwise commuting projectors,
which in view of Proposition 2.3.11 has the abstract structure of aσ-algebra, a
quantum stateρreduces thereon to a standard probability measure. In this sense the
“quantum probability” we are considering extends the classical notion. Differences
show up just when one deals with conditional probability involving incompatible
elementary observables.

An interesting case of (c) in the remark above is a convex combination of states
induced by unit vectors as in (2.79), where〈ψk,ψh〉=δhk,

ρ=

∑n

k=1

pkρψk.

By direct inspection, completing the finite orthonormal system{ψk}k=1,...,nto a
full Hilbertian basis ofH, one quickly proves that, defining

T=

∑n

k=1

pk〈ψk,〉ψk (2.80)

ρ(P) can be computed as

ρ(P)=tr(TP) P∈L(H)

In particular it turns out thatTis inB 1 (H), satisfiesT≥0 (so it is selfadjoint
for (3) in exercise 2.2.31) andtr T = 1. As a matter of fact, (2.80) is just the
spectral decomposition ofT, whose spectrum is{pk}k=1,...,n. This result is general
[5; 6]